By means of some algebraic techniques,especially the Binet-Cauchy formula,an explicit multi-soliton solution of the derivative nonlinear Schrdinger equation with vanishing boundary condition is attained based on a pur...By means of some algebraic techniques,especially the Binet-Cauchy formula,an explicit multi-soliton solution of the derivative nonlinear Schrdinger equation with vanishing boundary condition is attained based on a pure Marchenko formalism without needing the usual scattering data except for given N simple poles.The one-and two-soliton solutions are given as two special examples in illustration of the general formula of multi-soliton solution.Their effectiveness and equivalence to other approaches are also demonstrated.Meanwhile,the asymptotic behavior of the multi-soliton solution is discussed in detail.It is shown that the N-soliton solution can be viewed as a summation of N one-soliton solutions with a definite displacement and phase shift of each soliton in the whole process(from t→∞to t→+∞)of the elastic collisions.展开更多
An H^1-Galerkin mixed finite element method is discussed for a class of second order SchrSdinger equation. Optimal error estimates of semidiscrete schemes are derived for problems in one space dimension. At the same t...An H^1-Galerkin mixed finite element method is discussed for a class of second order SchrSdinger equation. Optimal error estimates of semidiscrete schemes are derived for problems in one space dimension. At the same time, optimal error estimates are derived for fully discrete schemes. And it is showed that the H1-Galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the LBB consistency condition.展开更多
We use the Fokas method to analyze the derivative nonlinear Schrodinger （DNLS） equation iqt （x, t） = -qxx （x, t）＋（rq^2）x on the interval [0, L]. Assuming that the solution q（x, t） exists, we show that it ca...We use the Fokas method to analyze the derivative nonlinear Schrodinger （DNLS） equation iqt （x, t） = -qxx （x, t）＋（rq^2）x on the interval [0, L]. Assuming that the solution q（x, t） exists, we show that it can be represented in terms of the solution of a matrix Riemann- Hilbert problem formulated in the plane of the complex spectral parameter ξ. This problem has explicit （x, t） dependence, and it has jumps across {ξ∈C｜Imξ^4 = 0}. The relevant jump matrices are explicitely given in terms of the spectral functions {a（ξ）, b（ξ）}, {A（ξ）, B（ξ）}, and {A（ξ）, B（ξ）}, which in turn are defined in terms of the initial data q0（x） = q（x, 0）, the bound- ary data g0（t）= q（0, t）, g1（t） = qx（0, t）, and another boundary values f0（t） = q（L, t）, f1（t） = qx（L, t）. The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.展开更多
This paper is concerned with the inverse scattering problems for Schrdinger equations with compactly supported potentials.For purpose of reconstructing the support of the potential,we derive a factorization of the sca...This paper is concerned with the inverse scattering problems for Schrdinger equations with compactly supported potentials.For purpose of reconstructing the support of the potential,we derive a factorization of the scattering amplitude operator A and prove that the ranges of （A＊ A） ^1/4 and G which maps more general incident fields than plane waves into the scattering amplitude coincide.As an application we characterize the support of the potential using only the spectral data of the operator A.展开更多
The present paper studies the unstable nonlinear Schr¨odinger equations, describing the time evolution of disturbances in marginally stable or unstable media. More precisely, the unstable nonlinear Schr¨odin...The present paper studies the unstable nonlinear Schr¨odinger equations, describing the time evolution of disturbances in marginally stable or unstable media. More precisely, the unstable nonlinear Schr¨odinger equation and its modified form are analytically solved using two efficient distinct techniques, known as the modified Kudraysov method and the sine-Gordon expansion approach. As a result, a wide range of new exact traveling wave solutions for the unstable nonlinear Schr¨odinger equation and its modified form are formally obtained.展开更多
In this paper, we study blow-up solutions of the Cauchy problem to the L2 critical nonlinear Schrdinger equation with a Stark potential. Using the variational characterization of the ground state for nonlinear Schrdin...In this paper, we study blow-up solutions of the Cauchy problem to the L2 critical nonlinear Schrdinger equation with a Stark potential. Using the variational characterization of the ground state for nonlinear Schrdinger equation without any potential, we obtain some concentration properties of blow-up solutions, including that the origin is the blow-up point of the radial blow-up solutions, the phenomenon of L2-concentration and rate of L2-concentration of blow-up solutions.展开更多
We consider the following nonlinear Schroodinger equations -ε^2△u ＋ u = Q（x）｜u｜^p-2u in R^N, u ∈ H^1（R^N）,where ε is a small positive parameter, N ≥ 2, 2 〈 p 〈 ∞ for N = 2 and 2 〈 p 〈2N/N-2 for N ≥ 3...We consider the following nonlinear Schroodinger equations -ε^2△u ＋ u = Q（x）｜u｜^p-2u in R^N, u ∈ H^1（R^N）,where ε is a small positive parameter, N ≥ 2, 2 〈 p 〈 ∞ for N = 2 and 2 〈 p 〈2N/N-2 for N ≥ 3. We prove that this problem has sign-changing（nodal） semi-classical bound states with clustered spikes for sufficiently small ε under some additional conditions on Q（x）.Moreover, the number of this type of solutions will go to infinity as ε→ 0^＋.展开更多
In this note, it is shown that the revision of the Kaup-Newell＇s works on 1ST for DNLS equation is only available in the ease of solving the bright one-soliton solution to the equation. An example is taken to illustr...In this note, it is shown that the revision of the Kaup-Newell＇s works on 1ST for DNLS equation is only available in the ease of solving the bright one-soliton solution to the equation. An example is taken to illustrate our point of view.展开更多
By Taylor expansion of Darboux matrix, a new generalized Darboux transformations(DTs) for a(2 + 1)-dimensional nonlinear Schrdinger(NLS) equation is derived, which can be reduced to two(1 + 1)-dimensional equation:a m...By Taylor expansion of Darboux matrix, a new generalized Darboux transformations(DTs) for a(2 + 1)-dimensional nonlinear Schrdinger(NLS) equation is derived, which can be reduced to two(1 + 1)-dimensional equation:a modified KdV equation and an NLS equation. With the help of symbolic computation, some higher-order rational solutions and rogue wave(RW) solutions are constructed by its(1, N-1)-fold DTs according to determinants. From the dynamic behavior of these rogue waves discussed under some selected parameters, we find that the RWs and solitons are demonstrated some interesting structures including the triangle, pentagon, heptagon profiles, etc. Furthermore, we find that the wave structure can be changed from the higher-order RWs into higher-order rational solitons by modulating the main free parameter. These results may give an explanation and prediction for the corresponding dynamical phenomena in some physically relevant systems.展开更多
In this paper, we give the Lax pair and construct the Darboux transformation of the Kundu-DNLS equation. Furthermore, the rogue wave solutions of the Kundu-DNLS equation are derived by using the Taylor expansion of th...In this paper, we give the Lax pair and construct the Darboux transformation of the Kundu-DNLS equation. Furthermore, the rogue wave solutions of the Kundu-DNLS equation are derived by using the Taylor expansion of the breather solution. What's more, the triangular and the circular patterns of the third rouge solution are displayed.展开更多
We proposed a simple potential harmonic(PH) scheme for calculating the non\|relativistic radial correlation energies of atomic systems. The scheme was applied to the low\|lying \%n\%\+1\%S\%(\%n\%=1,2) and \%n\%\+3\%...We proposed a simple potential harmonic(PH) scheme for calculating the non\|relativistic radial correlation energies of atomic systems. The scheme was applied to the low\|lying \%n\%\+1\%S\%(\%n\%=1,2) and \%n\%\+3\%S\%(\%n\%=2,3) states of the helium atom. The results exhibit a very stable convergence characterization in both the angular and radial directions with PH and generalized Laguerre functions(GLF) respectively, even though the method is non\|variational one. The ninth significant figure of the non\|relativistic radial energy(NRE) calculated for the ground state exactly agrees with that of the most accurate literature data from the modified configuration interaction method. The convergent NRE′s for the excited states 2\+1\%S\%, 2\+3\%S\% and 3\+3\%S\% with the similar accuracy were also obtained.展开更多
With a special gauge transformation,the Lax pair of the derivative nonlinear Shcrdinger (DNLS) equation turns to depend on the squared parameter λ = k2instead of the usual spec-tral parameter k. By introducing a new ...With a special gauge transformation,the Lax pair of the derivative nonlinear Shcrdinger (DNLS) equation turns to depend on the squared parameter λ = k2instead of the usual spec-tral parameter k. By introducing a new direct product of Jost solu-tions,the complete Hamiltonian theory of the DNLS equation is constructed on the basis of the squared spectral parameter,which shows that the integrability completeness is still preserved. This result will be beneficial to the further study of the DNLS equation,such as the direct perturbation method.展开更多
In this paper, we introduce and propose exact and explicit analytical solutions to a novel model of the nonlinear Schr¨odinger(NLS) equation. This model is derived as the equation governing the dynamics of modula...In this paper, we introduce and propose exact and explicit analytical solutions to a novel model of the nonlinear Schr¨odinger(NLS) equation. This model is derived as the equation governing the dynamics of modulated cutoff waves in a discrete nonlinear electrical lattice. It is characterized by the addition of two terms that involve time derivatives to the classical equation. Through those terms, our model is also tantamount to a generalized NLS equation with saturable;which suggests that the discrete electrical transmission lines can potentially be used to experimentally investigate wave propagation in media that are modeled by such type of nonlinearity. We demonstrate that the new terms can enlarge considerably the forms of the solutions as compared to similar NLS-type equations. Sine–Gordon expansion-method is used to derive numerous kink, antikink, dark, and bright soliton solutions.展开更多
A systematic method which is based on the classical Lie group reduction is used to find the novel exact solution of the cubic-quintic nonlinear Schrdinger equation (CQNLS) with varying dispersion,nonlinearity,and gain...A systematic method which is based on the classical Lie group reduction is used to find the novel exact solution of the cubic-quintic nonlinear Schrdinger equation (CQNLS) with varying dispersion,nonlinearity,and gain or absorption.Algebraic solitary-wave as well as kink-type solutions in three kinds of optical fibers represented by coefficient varying CQNLS equations are studied in detail.Some new exact solutions of optical solitary wave with a simple analytic form in these models are presented.Appropriate solitary wave solutions are applied to discuss soliton propagation in optical fibres,and the amplification and compression of pulses in optical fibre amplifiers.展开更多
We construct analytical self-similar solutions for the generalized (3+1)-dimensional nonlinear Schrdinger equation with polynomial nonlinearity of arbitrary order. As an example, we list self-similar solutions of quin...We construct analytical self-similar solutions for the generalized (3+1)-dimensional nonlinear Schrdinger equation with polynomial nonlinearity of arbitrary order. As an example, we list self-similar solutions of quintic nonlinear Schrdinger equation with distributed dispersion and distributed linear gain, including bright similariton solution, fractional and combined Jacobian elliptic function solutions. Moreover, we discuss self-similar evolutional dynamic behaviors of these solutions in the dispersion decreasing fiber and the periodic distributed amplification system.展开更多
A class of analytical solitary-wave solutions to the generalized nonautonomous cubic–quintic nonlinear Schrdinger equation with time-and space-modulated coefficients and potentials are constructed using the similarit...A class of analytical solitary-wave solutions to the generalized nonautonomous cubic–quintic nonlinear Schrdinger equation with time-and space-modulated coefficients and potentials are constructed using the similarity transformation technique. Constraints for the dispersion coefficient, the cubic and quintic nonlinearities, the external potential, and the gain （loss） coefficient are presented at the same time. Various shapes of analytical solitary-wave solutions which have important applications of physical interest are studied in detail, such as the solutions in Feshbach resonance management with harmonic potentials, Faraday-type waves in the optical lattice potentials, and localized solutions supported by the Gaussian-shaped nonlinearity. The stability analysis of the solutions is discussed numerically.展开更多
基金Supported by the National Natural Science Foundation of China(10775105)
文摘By means of some algebraic techniques,especially the Binet-Cauchy formula,an explicit multi-soliton solution of the derivative nonlinear Schrdinger equation with vanishing boundary condition is attained based on a pure Marchenko formalism without needing the usual scattering data except for given N simple poles.The one-and two-soliton solutions are given as two special examples in illustration of the general formula of multi-soliton solution.Their effectiveness and equivalence to other approaches are also demonstrated.Meanwhile,the asymptotic behavior of the multi-soliton solution is discussed in detail.It is shown that the N-soliton solution can be viewed as a summation of N one-soliton solutions with a definite displacement and phase shift of each soliton in the whole process(from t→∞to t→+∞)of the elastic collisions.
基金Supported by the National Natural Science Foundation of China (10601022)Natural Science Foundation of Inner Mongolia Autonomous Region (200607010106)Youth Science Foundation of Inner Mongolia University(ND0702)
文摘An H^1-Galerkin mixed finite element method is discussed for a class of second order SchrSdinger equation. Optimal error estimates of semidiscrete schemes are derived for problems in one space dimension. At the same time, optimal error estimates are derived for fully discrete schemes. And it is showed that the H1-Galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the LBB consistency condition.
基金supported by grants from the National Science Foundation of China (10971031 11271079+2 种基金 11075055)Doctoral Programs Foundation of the Ministry of Education of Chinathe Shanghai Shuguang Tracking Project (08GG01)
文摘We use the Fokas method to analyze the derivative nonlinear Schrodinger （DNLS） equation iqt （x, t） = -qxx （x, t）＋（rq^2）x on the interval [0, L]. Assuming that the solution q（x, t） exists, we show that it can be represented in terms of the solution of a matrix Riemann- Hilbert problem formulated in the plane of the complex spectral parameter ξ. This problem has explicit （x, t） dependence, and it has jumps across {ξ∈C｜Imξ^4 = 0}. The relevant jump matrices are explicitely given in terms of the spectral functions {a（ξ）, b（ξ）}, {A（ξ）, B（ξ）}, and {A（ξ）, B（ξ）}, which in turn are defined in terms of the initial data q0（x） = q（x, 0）, the bound- ary data g0（t）= q（0, t）, g1（t） = qx（0, t）, and another boundary values f0（t） = q（L, t）, f1（t） = qx（L, t）. The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.
基金The Major State Basic Research Development Program Grant (2005CB321701)the Heilongjiang Education Committee Grant (11551364) of China
文摘This paper is concerned with the inverse scattering problems for Schrdinger equations with compactly supported potentials.For purpose of reconstructing the support of the potential,we derive a factorization of the scattering amplitude operator A and prove that the ranges of （A＊ A） ^1/4 and G which maps more general incident fields than plane waves into the scattering amplitude coincide.As an application we characterize the support of the potential using only the spectral data of the operator A.
文摘The present paper studies the unstable nonlinear Schr¨odinger equations, describing the time evolution of disturbances in marginally stable or unstable media. More precisely, the unstable nonlinear Schr¨odinger equation and its modified form are analytically solved using two efficient distinct techniques, known as the modified Kudraysov method and the sine-Gordon expansion approach. As a result, a wide range of new exact traveling wave solutions for the unstable nonlinear Schr¨odinger equation and its modified form are formally obtained.
基金supported by National Science Foundation of China (11071177)
文摘In this paper, we study blow-up solutions of the Cauchy problem to the L2 critical nonlinear Schrdinger equation with a Stark potential. Using the variational characterization of the ground state for nonlinear Schrdinger equation without any potential, we obtain some concentration properties of blow-up solutions, including that the origin is the blow-up point of the radial blow-up solutions, the phenomenon of L2-concentration and rate of L2-concentration of blow-up solutions.
基金supported by NSFC(11301204)supported by program for outstanding young Technology Innovative team in universities of Hubei Province(T2014212)
文摘We consider the following nonlinear Schroodinger equations -ε^2△u ＋ u = Q（x）｜u｜^p-2u in R^N, u ∈ H^1（R^N）,where ε is a small positive parameter, N ≥ 2, 2 〈 p 〈 ∞ for N = 2 and 2 〈 p 〈2N/N-2 for N ≥ 3. We prove that this problem has sign-changing（nodal） semi-classical bound states with clustered spikes for sufficiently small ε under some additional conditions on Q（x）.Moreover, the number of this type of solutions will go to infinity as ε→ 0^＋.
基金the Postdoctoral Fund of Huazhong University of Science and Technology under Grant No.0128011006
文摘In this note, it is shown that the revision of the Kaup-Newell＇s works on 1ST for DNLS equation is only available in the ease of solving the bright one-soliton solution to the equation. An example is taken to illustrate our point of view.
基金Supported by National Natural Science Foundation of China under Grant Nos.11775121 and 11435005K.C.Wong Magna Fund in Ningbo University
文摘By Taylor expansion of Darboux matrix, a new generalized Darboux transformations(DTs) for a(2 + 1)-dimensional nonlinear Schrdinger(NLS) equation is derived, which can be reduced to two(1 + 1)-dimensional equation:a modified KdV equation and an NLS equation. With the help of symbolic computation, some higher-order rational solutions and rogue wave(RW) solutions are constructed by its(1, N-1)-fold DTs according to determinants. From the dynamic behavior of these rogue waves discussed under some selected parameters, we find that the RWs and solitons are demonstrated some interesting structures including the triangle, pentagon, heptagon profiles, etc. Furthermore, we find that the wave structure can be changed from the higher-order RWs into higher-order rational solitons by modulating the main free parameter. These results may give an explanation and prediction for the corresponding dynamical phenomena in some physically relevant systems.
文摘In this paper, we give the Lax pair and construct the Darboux transformation of the Kundu-DNLS equation. Furthermore, the rogue wave solutions of the Kundu-DNLS equation are derived by using the Taylor expansion of the breather solution. What's more, the triangular and the circular patterns of the third rouge solution are displayed.
基金Supported by the National Natural Science Foundation of China(No. 2 970 30 0 3)
文摘We proposed a simple potential harmonic(PH) scheme for calculating the non\|relativistic radial correlation energies of atomic systems. The scheme was applied to the low\|lying \%n\%\+1\%S\%(\%n\%=1,2) and \%n\%\+3\%S\%(\%n\%=2,3) states of the helium atom. The results exhibit a very stable convergence characterization in both the angular and radial directions with PH and generalized Laguerre functions(GLF) respectively, even though the method is non\|variational one. The ninth significant figure of the non\|relativistic radial energy(NRE) calculated for the ground state exactly agrees with that of the most accurate literature data from the modified configuration interaction method. The convergent NRE′s for the excited states 2\+1\%S\%, 2\+3\%S\% and 3\+3\%S\% with the similar accuracy were also obtained.
基金Supported by the National Natural Science Foundation of China (10705022)
文摘With a special gauge transformation,the Lax pair of the derivative nonlinear Shcrdinger (DNLS) equation turns to depend on the squared parameter λ = k2instead of the usual spec-tral parameter k. By introducing a new direct product of Jost solu-tions,the complete Hamiltonian theory of the DNLS equation is constructed on the basis of the squared spectral parameter,which shows that the integrability completeness is still preserved. This result will be beneficial to the further study of the DNLS equation,such as the direct perturbation method.
文摘In this paper, we introduce and propose exact and explicit analytical solutions to a novel model of the nonlinear Schr¨odinger(NLS) equation. This model is derived as the equation governing the dynamics of modulated cutoff waves in a discrete nonlinear electrical lattice. It is characterized by the addition of two terms that involve time derivatives to the classical equation. Through those terms, our model is also tantamount to a generalized NLS equation with saturable;which suggests that the discrete electrical transmission lines can potentially be used to experimentally investigate wave propagation in media that are modeled by such type of nonlinearity. We demonstrate that the new terms can enlarge considerably the forms of the solutions as compared to similar NLS-type equations. Sine–Gordon expansion-method is used to derive numerous kink, antikink, dark, and bright soliton solutions.
基金Supported by the National Natural Science Foundation of China under Grant No.10735030the K.C.Wong Magna Fund in Ningbo University and the Scientific Research Foundation of Graduate School of Ningbo University
文摘A systematic method which is based on the classical Lie group reduction is used to find the novel exact solution of the cubic-quintic nonlinear Schrdinger equation (CQNLS) with varying dispersion,nonlinearity,and gain or absorption.Algebraic solitary-wave as well as kink-type solutions in three kinds of optical fibers represented by coefficient varying CQNLS equations are studied in detail.Some new exact solutions of optical solitary wave with a simple analytic form in these models are presented.Appropriate solitary wave solutions are applied to discuss soliton propagation in optical fibres,and the amplification and compression of pulses in optical fibre amplifiers.
基金Supported by the Applied Nonlinear Science and Technology from the Most Important Among all the Top Priority Disciplines of Zhejiang Province
文摘We construct analytical self-similar solutions for the generalized (3+1)-dimensional nonlinear Schrdinger equation with polynomial nonlinearity of arbitrary order. As an example, we list self-similar solutions of quintic nonlinear Schrdinger equation with distributed dispersion and distributed linear gain, including bright similariton solution, fractional and combined Jacobian elliptic function solutions. Moreover, we discuss self-similar evolutional dynamic behaviors of these solutions in the dispersion decreasing fiber and the periodic distributed amplification system.
基金Project supported by the National Natural Science Foundation of China(Grant No.11175158)the Natural Science Foundation of Zhejiang Province of China(Grant No.LY12A04001)
文摘A class of analytical solitary-wave solutions to the generalized nonautonomous cubic–quintic nonlinear Schrdinger equation with time-and space-modulated coefficients and potentials are constructed using the similarity transformation technique. Constraints for the dispersion coefficient, the cubic and quintic nonlinearities, the external potential, and the gain （loss） coefficient are presented at the same time. Various shapes of analytical solitary-wave solutions which have important applications of physical interest are studied in detail, such as the solutions in Feshbach resonance management with harmonic potentials, Faraday-type waves in the optical lattice potentials, and localized solutions supported by the Gaussian-shaped nonlinearity. The stability analysis of the solutions is discussed numerically.
